I can't tell you how this is supposed to work in Turbotax because I haven't tested the interview. Someone else might be able to help. This is what's supposed to happen if you were doing it manually.
First, all your interest on the earlier loan is deductible, assuming it was all acquisition debt.
Then, on the refinanced loan, let's assume the entire $584K original balance was acquisition debt, and you had $4000 in closing costs and $5000 applied to your escrow account. (Your acquisition debt is $588K and your equity debt is $5K.) In that case, your starting loan balance is 99.15% acquisition debt so your interest is 99.15% deductible (588/593). Then you look at your loan balance at the end of the year. If it is below $588K, then your loan is 100% deductible at the end of the year. Average first month and last month, ((99.15+100)/2) and your interest should be 99.58% deductible for this year.
I believe you can also calculate each months' deductible percentage separately, but the IRS provides the first/last average method as simpler. There's also a worksheet here along with the rules https://www.irs.gov/pub/irs-pdf/p936.pdf
(And if the earlier 584K loan was also a refinance and included some equity debt, you will have to go back to your original purchase to figure out how much acquisition debt you really have in the loan.)
I can't tell whether you did the worksheet wrong or whether Turbotax is wrong, because I haven't tested that part of the program. Someone else might be able to help. But that's the answer you should be getting.
*Answers are correct to the best of my ability at the time of posting but do not constitute legal or tax advice.*
